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Natural Numbers
Published in Nita H. Shah, Vishnuprasad D. Thakkar, Journey from Natural Numbers to Complex Numbers, 2020
Nita H. Shah, Vishnuprasad D. Thakkar
First principle of mathematical induction: If a statement P(n) for natural numbers is true for n = 1 and P(k) is true⇒P(k+1) is true, then the statement is true for all natural numbers.
Numbers, trigonometric functions and coordinate geometry
Published in Alan Jeffrey, Mathematics, 2004
Mathematical propositions often involve some fixed integer n, say, in a special role, and it is desirable to infer the form taken by the proposition for arbitrary integral n from the form taken by it for the specific value n = n1. The logical method by which the proof of the general proposition, if true, may be established, is based on the properties of natural numbers and is called mathematical induction.
Preliminaries
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
which again corresponds to the second De Morgan’s Law: A∼(p1∨p2∨…∨pN)⇔(∼p1∧∼p2∧…∧∼pN) $$ A\sim (p_1\vee p_2\vee \ldots \vee p_N)\quad \Leftrightarrow \quad (\sim p_1\wedge \sim p_2 \wedge \ldots \wedge \sim p_N) $$ Principle of Mathematical Induction. Using the proof-by-contradiction concept and the negation rules for quantifiers, we can easily prove the Principle of Mathematical Induction. Let T(n) be an open statement for n∈N $ n\in \mathbb N $ . Suppose that: 1.T(1)(is true)2.T(k)⇒T(k+1)∀k∈N $$ \begin{array}{ll} \text{1.}\quad&T(1) \text{(is} \text{ true)}\\ \text{2.}&T(k) \Rightarrow T(k+1)\quad \forall k\in \mathbb N \end{array} $$
Semi-analytical solutions of shallow water waves with idealised bottom topographies
Published in Geophysical & Astrophysical Fluid Dynamics, 2023
This is proven via mathematical induction by examining the recursion relationships for u, v, and h in (5). Condition (18a) is demonstrated by examining the following relationships Therefore, when n = 0, equations (19a–c) representing the relationship between the initial and first components for u, v, and h become Employing (17a–c) it can be shown that equations (20a–c) reduce to the following relationship and continuing this argument for yields equation (18a). Following similar arguments yields (18b).
The problem and geometric application of infinite sequences formed from three given numbers by calculating their pairwise means
Published in International Journal of Mathematical Education in Science and Technology, 2022
In this study, we present activities related to the sides of triangles. The activities are carried out in several stages and demonstrate the use of various tools, conventional (compass, straightedge and pencil) and computerized (Excel, GeoGebra). They require knowledge from various fields of mathematics: sequences and limits, recursion, geometric progression, mathematical induction and triangle geometry. The activities are intended for both pre-service and experienced teachers who are studying in teaching colleges and aim to enrich and deepen the teachers’ knowledge and skills regarding limits in geometry.
Event-based state estimation for time-varying stochastic coupling networks with missing measurements under uncertain occurrence probabilities
Published in International Journal of General Systems, 2018
Hongxu Zhang, Jun Hu, Lei Zou, Xiaoyang Yu, Zhihui Wu
Proof To obtain the recursion of upper bound , we employ the mathematical induction approach. Note that , then we need to show that under the assumption that .